Abstract

We assume that the manifold with boundary, $X,$ has a Spin${}_{\mathbb{C}}$-structure with spinor bundle $S\mspace{-10mu}/$. Along the boundary, this structure agrees with the structure defined by an infinite order, integrable, almost complex structure and the metric is Kähler. In this case the Spin${}_{\mathbb{C}}$-Dirac operator $\eth$ agrees with $\bar{\partial}_b + \bar{\partial}_b^*$ along the boundary. The induced CR-structure on $bX$ is integrable and either strictly pseudoconvex or strictly pseudoconcave. We assume that $E\to X$ is a complex vector bundle, which has an infinite order, integrable, complex structure along $bX,$ compatible with that defined along $bX.$ In this paper we use boundary layer methods to prove subelliptic estimates for the twisted Spin${}_{\mathbb{C}}$-Dirac operator acting on sections on $S\mspace{-10mu}/ \otimes E$. We use boundary conditions that are modifications of the classical $\bar{\partial}$-Neumann condition. These results are proved by using the extended Heisenberg calculus.